Full Title --
Part 4 of 4: The Heart and Soul of Marxian Theory --
A Fundamental Inquiry into the Nature of Human Social Productive Force, and of Its Marxian Conception.
Operationalization --
Step-by-Step Construction of an ‘Historically-Generic’, ‘Qualo-Quantitative’ Measure of Human-Social Productive Force that is Adequate to its Concept.
Dear Readers,
This blog-entry contains part of my “improvement” of one more text
by the E.A.g. [Equitist Advocacy
group], this one entitled “The Heart and Soul of
Marxian Theory”.
I have already provided my version of their text, in the first two parts. I planned to then add two new sections, as a third and fourth part -- the present blog-entry forming the fourth and final part in this series of blog-entries.
The final two parts are intended to set forth some of the fruits of the latest research by the Foundation regarding the critical, immanent ‘extention’ of the central Marxian concept -- the concept of "the social forces of production" -- to the concept of the ‘‘‘human-societal [self-]force of human-societal [self-]reproduction’’’, in part via an immanent critique of the ideology-compromised science of Darwinian biology, resulting in F.E.D.’s theory of ‘Dialectical Meta-Darwinism’ as a positive fruition of that immanent critique.
Here are links to my three earlier blog-entries here, corresponding to the three earlier parts --
http://capitalismsfundamentalflaw-wayforward.blogspot.com/2013/05/part-1-of-4-heart-and-soul-of-marxian.html
http://capitalismsfundamentalflaw-wayforward.blogspot.com/2013/05/part-2-of-4-heart-and-soul-of-marxian.html
http://capitalismsfundamentalflaw-wayforward.blogspot.com/2013/12/part-3-of-4-heart-and-soul-of-marxian.html
Regards,
Miguel
Miguel
Introduction to Part 4. As the upshot of
the three preceding Parts, we have seen the evidence, from across the entire
progression of Marx’s writings, of the centrality of the concept that he named
‘the social forces of production’, in the Marxian theory of human social
evolution/revolution.
Also, we have arrived, as of the conclusion of Part 3, at a
fundamental formulation of the nature of human social productive force -- that
it refers to sustained
accelerations in the rate
of human societal self-reproduction, to the sustained, self-accelerated self-expansion of ‘human socio-mass’,
which includes both living human biomass, and in-use human
artifacts-mass, since both may contribute to the ‘‘‘forcefulness’’’ / sustained accelerated growth rate
of ‘human socio-mass’.
The questions central to this Part are -- ¿How can we best quantify
the ‘self-productive force’ of any given human society, and of the planetary
human species as a whole?
-- and -- ¿How can we best qualify
this quantification
in terms of the best [qualitative]
unit of measure, the best ‘measuremental’ “dimension”, of this quantification, e.g., in terms of [one,
or of combinations of some to all of] the hour
unit, the ton unit,
the mile unit, the dollar unit, the percent [%] unit,
etc.?
¿In terms of what [qualitative] units-of-measure are we to count the magnitude, to aggregate the ‘‘‘quantifiers’’’, of human societal productive force?
¿In terms of what [qualitative] units-of-measure are we to count the magnitude, to aggregate the ‘‘‘quantifiers’’’, of human societal productive force?
The purpose of this fourth and final Part is to present a
step-by-step construction, starting from a simplest first approximation, of a
historically-generic way of
measuring, at any
time in human history, the magnitude
of that ‘force of human societal self-reproduction’, the magnitude of the
acceleration in each historically-specific society’s self-caused reproduction
of itself, as a ‘human genome/human phenome unity’, a way that applies equally
to each of the qualitatively different, ‘socio-ontologically distinct’ social
forms -- to the social formations, dominant social relations of production, and
‘‘‘modes of [human social re]production’’’ -- displayed in human
history on this planet, and one which applies equally also to the expected
future new forms of human society that we are capable of envisioning.
A review, by the reader, of the first three Parts would be
helpful to the reader, at this juncture, as, below, we build, step-by-step, layer-by-layer, toward
the most advanced form of such a way of measuring this ‘self-force’
of human society that we will present herein, based upon the steps in the
formation of our conception of this ‘self-force’ in those previous Parts, and,
especially, in Part three.
Step-by-Step
Construction of a Human-Society Self-Reproductive Self-Force ‘Quanto-Qualitative’ Metric. Let us begin with a first step that considers
ways to measure increases in the ‘‘‘velocity’’’ of human social production at
the ‘‘‘micro-scale’’’ as identified in Part 3, and then work our way up from
there, through the ‘‘‘meso-scale’’’ of human social production, all the way up
to the ‘‘‘macro-scale’’’ of the global human species, the scale of ‘planetized
humanity’ as a whole, critiquing each metric option we encounter and consider
in this progression of metric options in terms of the degree to which each such
metric option satisfies the concept of this ‘human society self-reproducing
self-force of human society’ to which we
arrived in Part 3.
a. Micro-scale. Metrics for
the ‘‘‘productive force’’’
of single, local instrumentalities of production. Consider the options for measuring the
acceleration of the output-rate of one single, particular kind of good by an
upgrade in (1) an individual
instrument, e.g., an individual, hand-held tool, wielded by an individual
worker, or in (2) an
individual machine, in its productive use, by a human worker, or by a team of
human workers, or by a factory-scale “automatic system of machinery” [Marx], or
in (3) an
individual worker, or in a team of workers, by incremental training, producing,
in them, incremental production skills, or in (4) combinations of the preceding cases -- of both upgraded instrumentalities and upgraded worker(s’) skill(s) -- of increases
in the “technical composition” of labor-power [Marx], of “living labor” [Marx],
combined with increases in the “technical composition” [Marx] of fixed capital
plant and equipment, of “dead labor” [Marx], e.g., as any of these scenarios
might take place in a local operating unit, e.g., in a single “plant”, or a
single “factory”, of in a single “individual capital”. In Part 3 we explicitly considered three options --
- Such increases in the rate of production of an individual kind of good might be measured as the number of discrete units of that particular good that is being finished, and "put-out", or "out-put", per unit of time -- say per the hour unit of time-measurement, i.e., in “#/hr.” as a compound metrical unit qualifier;
- Or, such increases in the rate of production of an individual kind of good might be measured as the number of units of the physical mass of a particular good that is being finished/output, say in ton units, e.g., per hour unit, i.e., in “tons/hr.” as a compound metrical unit qualifier;
- Or, such increases in the rate of production of an individual kind of good might be measured as the number of units of money, e.g., in dollar units, of the unit sales price, or in the unit margin [profit], gross or net, of that particular good, that is being output per hour unit, i.e., in “$s/hr.” as a compound metrical unit qualifier.
The 3rd option above fails, for our purposes, because,
even if detailed records of the prices of commodities were available for past
human social formations in which the money-form of value, and the [money-circulated]
commodity-form of value, were, in fact, already extant, productive force
metric quantifiers qualified by units of “Xs/hr.”
[using 'X' as a generic currency unit qualifier] would nonetheless be inapplicable
to the [typically] earlier human social formations in which the money-relation,
and the [money-circulated] commodity-relation, had yet to emerge.
The 1st option above is useful, for the micro-scale,
and for a single category of product.
But it cannot be extended to the larger scales, because they are also multi-product-category
scales: counting the product “apples”
and the product “oranges” in the same cardinal units -- e.g., counting a unit
of output of an airliner product as 1, and a unit of
microchip output also as 1, asserting, in effect, that producing 1
airliner per hour is equivalent to producing 1 microchip per hour,
that “1 airliner = 1 microchip”, would be,
well, “mixing up apples and oranges”, and would therefore yield highly
misleading aggregate results.
The 2nd option above is of use in our local,
one-product scenario, and also holds out some hope of being “extendable” to
multi-product contexts, and to large geographical scales. E.g., one airliner obviously “weighs in” at a
lot more -- with a lot larger quantifier for its physical mass unit -- than does one
microchip. However, the sustained
acceleration of human societal, and of global human species, self-reproduction
is far from perfectly correlated with the increasing mass of product
output. Moreover, measuring aggregate
product mass growth-rate leaves out a central component of what human societal
self-reproductive force must mean -- a growing growth-rate of human population: “increased productivity, the increase of needs
. . . what is fundamental to both of these, the increase of population” [Marx and Engels, The German Ideology,
Progress Publishers, p. 43]; “The
community itself appears as the first great force of production” [Marx, «Grundrisse», Nicolaus, p. 495].
So
let us see how far we can get, toward a metric which does
justice to the concept of productive force at which we have arrived, and
let us also, by, for
now, skipping the measurement of our products altogether -- whether by
"census" count of discrete product units produced per hour, or by
counting mass units of product produced per hour, or by
counting monetary units of product "value" produced per hour, alike --
instead focusing on what we expect our consumption
of these, our product[ion]s, to [re]produce -- to sustain, and to expand -- our population.
b. Meso-scales. Metrics for
the ‘‘‘productive force’’’
of large but sub-national geographical
areas. Consider a census population count, p_{t}, a
representative value for the cardinal number of living human beings, alive
during discrete time-period t -- say during year
t -- as a
candidate “dimensionless” metric for a given human society’s self-reproductive
force, graphed, over successive values of t, in the space of p_{t} _|_ t, wherein '_|_' stands for the phrase 'is perpendicular to'.
Clearly, as we can see from the quotes from Marx’s and Engels’s writings
presented at the end of sub-section a., this metric is getting at something of what
Marx and Engels had in mind for their phrase “force of production”.
However, it fails to deliver the metric that we need by our
criterion of fair comparability: it does
not provide an adequate evaluation of societal reproductive force for the unbiased comparison
of one society to another, for either two contemporaneous societies on the
planet, or for a later versus an older such society.
It delivers only an “absolute” number, not a number
relativized to past values in the population levels time-series for that society.
It also delivers merely a ‘statical’ count: it does not directly and explicitly measure the population change dynamics
of that society, only a succession of periodic population levels, in which the changes between those time periods is left in mere implicitude.
In answer to these criticisms of p_{t} as productive force metric,
we can amend our metric to a more dynamical and more past-self-relative form,
as Dp_{t-1} = p_{t} - p_{t-1} [in which we use 'D' to denote "Delta", standing for the "forward finite Difference operator"], which counts a one year change in
population, yielding the representative number of live human individuals added to the population of the
society in question from year t -1
to year t.
However, note that this metric is still biased, toward giving
the initially larger population-size society higher marks for population
reproductive rate, still failing our criterion of unbiased comparability.
This same bias remains in place, even if we amend our metric
further, by shifting to the yet more dynamical, full Newtonian, “divided
differences” form, measured in metrical units of “people gained per year [net]”, '# of people/yr.
--
.....Dp_{t-1}.......=.....p_{t} - p_{t-1}.......=.........p_{t} - p_{t-1}..........=..........p_{t} - p_{t-1}......=.....Dp_{t-1}.
_____..............._______ ................________
..........Dt............=....t - (t-1).........=..................1
Therefore, let us, instead, test, against our concept of
societal productive force, and its qualifying criteria, a metric that assesses
the rate of growth of a society’s representative population, not in terms of the elementary
arithmetical operation of the subtraction
of most-recent, consecutive past year’s representative population level from the most current
year’s representative population level, but using the more advanced elementary arithmetical
operation of division:
..........p_{t}
..._____
........p_{t-1}
-- which answers the question: ¿How
many "times" greater,
in rational numbers terms, is the most recent year’s representative population level, relative
to the representative population level of the immediately, consecutively previous year, for the society in
question?
In moving to this amended, ‘divided totalities’,
ratio-metric, we have moved from the "Newtonian tradition" of dynamics
measurements, to the ‘‘‘econometric tradition’’’ of dynamics measurements.
That is, the annual “percentage growth rate of real GDP” [i.e., of “real” [“deflated”] Gross Domestic Product] of nation-state societies is measured, in current econometric practice, by the ratio --
That is, the annual “percentage growth rate of real GDP” [i.e., of “real” [“deflated”] Gross Domestic Product] of nation-state societies is measured, in current econometric practice, by the ratio --
...................rGDP_{t}
______________ - 1
................rGDP_{t-1}
-- which is a “net”
gain ratio for monetary-unit-aggregates-measured economic activity, i.e., one
which subtracts last year’s activity level from this year’s, in the numerator,
before comparing, using the division operation, the resulting, "net" rGDP_{t} gain, as numerator, to last year’s
economic activity level, as the denominator, thus combining the ‘‘‘subtractive’’’, or ‘‘‘Difference’’’,
approach, with the ‘‘‘division’’’, '/', approach to change measurement, to the measurement of dynamics, a fact which is more obvious in the
equivalent expression --
.rGDP_{t} - rGDP_{t-1}..........................................DrGDP_{t-1}
______________ .........=.........______________
................rGDP_{t-1}..............................................................rGDP_{t-1}
-- in which we see that the growth rate of a human social
economy, so measured, is greater than zero only if this year’s deflated GDP is greater than last year’s. Otherwise, if this year’s real GDP simply
matched last year’s, i.e., if rGDP_{t} = rGDP_{t-1 }-- a
zero real GDP growth
scenario -- then the “gross”
gain rate metric formula rGDP_{t}/rGDP_{t-1} would yield a “growth”
rate of 100%,
instead of the 0%
growth rate that the “net”
gain rate formula above yields for the same scenario.
Note that, p_{t}/p_{t-1}, a “past-self-relativized”,
or ‘‘‘past-self-normalized’’’,
“gross”
gain rate metric of the population growth rate of a given human society, is measured in
compound units of cardinal count over cardinal count, of count ÷ count --
of count/count, of count^{+1} x count^{-1}, of count^{+1}^{-1}, or of count^{0} --
#/# = # ÷ # = #^{+1} x #^{-1} = #^{+1}^{-1} = #^{0} = %
-- the “zero degree”, or “dimensionless”, “percentage” unit of measurement.
of count/count, of count^{+1} x count^{-1}, of count^{+1}^{-1}, or of count^{0} --
#/# = # ÷ # = #^{+1} x #^{-1} = #^{+1}^{-1} = #^{0} = %
-- the “zero degree”, or “dimensionless”, “percentage” unit of measurement.
Regardless of how much larger one human society is relative to
another, to start out with, this ratio-metric gives us a self-relative gain ratio, or gain rate, of the population level
of each society, in units of percents [% units], and is thus both dynamical, and more
comparable, across human societies on this planet.
Let us now follow suit even further with the ‘‘‘econometric
tradition’’’, and amend our “gross
gain ratio” metric to a “net
gain ratio” metric --
..p_{t} - p_{t-1}.......................................Dp_{t-1}
________ ...........=..........._________.
...........p_{t-1}.......................................................p_{t-1}
The ratio above compares, by division, a time-period, annual value, for year t -1, placed in the denominator, to a “between periods” difference-comparison result, created by a subtraction, i.e., to a value [possibly positive, zero, or negative] measuring the gain in population, from year t -1 to year t, placed in the numerator.
¿Can
we move
from the formula above, to one with a more numerator/denominator
“symmetric” comparison, more numerator/denominator “apples-to-apples”
comparison, of most recent annual human population gains to
consecutive-past,
immediate-past annual human population gains?
Yes.
And when we do so, we preserve the idea of comparing most
recent human population to immediately, consecutively previous human population
-- of comparing the present “self” of population to its just-past “self” -- in a
way which brings back aspects of the Newtonian, “divided differences”, ‘‘‘time-relative’’’ formula-form, and
reconciles it, in a special way, with the ‘‘‘econometric’’’, ‘past-self-relative’ formula-form
--
........p_{t} - p_{t-1}.........................................Dp_{t-1}
__________ ..........=..........._________.
.....p_{t-1} - p_{t-2}........................................Dp_{t-2}
The form[ula] to which we have just arrived above can be characterized as one which compares, by division, a difference-measure of most-recent time-period's population gain, [relative] to a difference-measure of the immediately-previous time-period's population gain.
In what Karl Seldon calls ‘time-offset self-space’, the ‘past-self-relative’, period-population gain-rate formula, above, Dp_{t-1}/Dp_{t-2}, coincides, in form, with the Newtonian, “divided differences”, “time rate of change”, population growth rate, or “population-difference divided by time-difference” formula, Dp_{t-1}/D(t-1), or Dp_{t}/Dt, but not, entirely, in content.
The ‘self-space’ graphical construct is often used, under other names, such as “graphical iteration”, to analyze, to present, and to “solve” especially the nonlinear , “fractal”, often “chaotic” finite difference equations, e.g., “quadratic maps” [see, for example, Heinz-Otto Peitgen, Harmut JÃ¼rgens, and Dietmar Saupe, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, [NY: 1992], p. 510 et passim].
Seldon’s ‘self-space’ names a graph-space in which the
vertical axis, say the Y axis, is assigned, in generic terms, to a time-series function, z_{t}, or, in our specific examples, to p_{t}, or to rGDP_{t}, but in which the
horizontal axis, say the X axis,
is not assigned to t, but, instead, to, generically, that same time-series function, but offset one time-period to the past, z_{t-1}, or, for our specific examples, to p_{t-1}
or to rGDP_{t-1}.
In the resulting --
Y _|_ X = z_{t} _|_ z_{t-1}
-- 'auto-space', or 'self-space', as distinct from the more conventional --
Y _|_ X = z_{t} _|_ t
-- 'allo-space', or 'hetero-space', of [discretized] Newtonian dynamics, the “ordered pairs” to be graphed in this graph-space are [if we indicate ordered pairs by means of their enclosure in "angle-brackets", e.g., by '< _, _ >] --
< x, y > = < z_{t-1}, z_{t} >
-- and not --
< x, y > = < t, z_{t} >.
Thus, the Newtonian-dynamical “differences” ratio, generically Dy/Dx = Dz_{t}/Dt, or, in the cases of our specific examples, Dp_{t}/Dt or DrGDP_{t}/Dt -- all precursors to the Newtonian dynamical “differential”, or “instantaneous rate of change with respect to time”, that results from “taking the limit” as Dx ---> 0 --
lim_{(}_{D}_{x }_{--->}_{ 0)}(DY(x)/Dx) = dz(t)/dt
-- become, instead, generically --
Dy/Dx = Dz_{t}/Dz_{t-1}
-- or, in the specific cases of our examples --
Dy/Dx = Dp_{t}/Dp_{t-1}
-- or --
Dy/Dx = DrGDP_{t}/DrGDP_{t-1}.
In the resulting --
Y _|_ X = z_{t} _|_ z_{t-1}
-- 'auto-space', or 'self-space', as distinct from the more conventional --
Y _|_ X = z_{t} _|_ t
-- 'allo-space', or 'hetero-space', of [discretized] Newtonian dynamics, the “ordered pairs” to be graphed in this graph-space are [if we indicate ordered pairs by means of their enclosure in "angle-brackets", e.g., by '< _, _ >] --
< x, y > = < z_{t-1}, z_{t} >
-- and not --
< x, y > = < t, z_{t} >.
Thus, the Newtonian-dynamical “differences” ratio, generically Dy/Dx = Dz_{t}/Dt, or, in the cases of our specific examples, Dp_{t}/Dt or DrGDP_{t}/Dt -- all precursors to the Newtonian dynamical “differential”, or “instantaneous rate of change with respect to time”, that results from “taking the limit” as Dx ---> 0 --
lim_{(}_{D}_{x }_{--->}_{ 0)}(DY(x)/Dx) = dz(t)/dt
-- become, instead, generically --
Dy/Dx = Dz_{t}/Dz_{t-1}
-- or, in the specific cases of our examples --
Dy/Dx = Dp_{t}/Dp_{t-1}
-- or --
Dy/Dx = DrGDP_{t}/DrGDP_{t-1}.
In this sense, we have reconciled/combined our "Newtonian
dynamics metric" and/with our "econometric dynamics metric", both for human population growth
rate dynamics as a proxy for the growth of the human social forces of production.
A generic rendering of this systematic dialectic of dynamics-measuring formulas -- a presentational dialectic that we have just traversed for the specific case of our developing dynamics-metric for social reproductive force -- is portrayed in the following image --
A generic rendering of this systematic dialectic of dynamics-measuring formulas -- a presentational dialectic that we have just traversed for the specific case of our developing dynamics-metric for social reproductive force -- is portrayed in the following image --
However, this proxy for human societal self-reproductive
force still tracks only human population growth rates of this special,
‘past-self-relative’ kind.
It tracks only the growth rate of human-Genomic living biomass.
It tracks only the growth rate of human-Genomic living biomass.
This proxy still lacks any assessment of the contribution of human
artifacts, of the objects which incarnate, materialize, and objectify the human Phenome, e.g., of the
human social means of social production.
Such a metric might be apt to measure the ‘species biomass
self-productivity’, or the Darwinian fitness, of a pre-human, nearly
'artifact-less', nearly ‘‘Phenome-less’ social-animal species, like,
say, dolphin.
But it will not do to capture the ‘Meta-Darwinian fitness’ of an ‘artifactful’, ‘Phenome-centered’ species like humanity!
But it will not do to capture the ‘Meta-Darwinian fitness’ of an ‘artifactful’, ‘Phenome-centered’ species like humanity!
In moving on and up to the highest scale that we plan to
address here, the macro-scale of the/a planetary human species as a whole, let us
see if we can extend our population-growth-rate-only metrical construct, to a
metrical construct which embraces also the necessarily concomitant, though not
at all necessarily proportionate, growing growth rate of human-artifacts-mass, of the
“non-living” embodiments of the human Phenome,
as well as the growing growth rate of the living embodiments of the
human Genome, which we have already achieved above.
c. Macro-scale.
Metrics for the ‘‘‘societal self-reproductive
force’’’ of the global[ized]
human species.
Near the end of the previous Part of this text, Part 3, we
stated that: “The challenge is this: Our
‘productive force’/‘Meta-Darwinian fitness’ metric needs two
components -- one quantifying the contribution of ‘the human Phenome’, the
other quantifying the contribution of “the human Genome”. And both of these components need to be
measured with a common unit of measure, i.e., via a common “dimension”.”
We propose to meet that challenge by defining a concept that we call
‘human socio-mass’, S_{t}, as the sum of two
components for each time-period,
t, in which S_{t}
is measured -- as the sum of (1)
the human biomass for time-period
t, i.e., the
physical mass of living human beings, to be measured in grams
[gm.], and denoted
by G_{t},
short for ‘human Genomic
[bio]mass for time-period t’,
and (2) the
physical mass of “live” human artifacts for time-period t, i.e., the mass of
human artifacts [that were] concurrently in working order, and [that were] currently in use in that time-period, also
to be measured in grams [gm.], and denoted by P_{t}, short for ‘human Phenomic mass for
time-period t’
--
S_{t }=
G_{t} + P_{t}
-- so that our ‘past-self-relative divided-differences self-space
finite-difference growth-rate ratio-metric’ for time-period t becomes --
............................................S_{t} - S_{t-1}.............................................DS_{t-1}
f_{t}..... ..........=.............__________ ..........=............_________.
.........................................S_{t-1} - S_{t-2}............................................DS_{t-2}
Note: The variables G_{t} and P_{t} are not defined for negative values.
We are thus measuring
the ‘self-reproductive
self-momentum’, or the ‘self-reproductive
self-velocity’, of a given social unit -- and, by comparing it, by division, against the past-self-relative rate of change of such ‘self-momenta’,
or self-velocities’, also measuring its ‘self-reproductive self-force’ -- by measuring the ‘past-self-relative’,
‘past-period-relative’ gain/growth of the numerator's ‘self-velocity’, or 'self-momentum', of ‘human socio-mass’ net self-expansion, as rendered in the image below, near the bottom of this blog-entry.
In that image, the ‘^{o}’
‘‘‘headdresses’’’ of certain symbols identify 'quantifiable metrical qualifiers' operators, ‘qualo-quantitative’,
or ‘qualo-quantitative’, values
[after the "monadic" symbology of Diophantus, «Arithmetica»,
circa 250 C.E.], whereas mere bold-face algebraic letter-symbols, without such ‘‘‘headdresses’’’, identify merely quantitative values, e.g., "Real" number values, mere ‘‘‘quantifiers’’’.
We say that positive[-signed], ‘human
socio-mass’ increasing societal ‘[self-re]productive self-velocity’, or ‘self-re]productive self-momentum’, exists, for time-period t,
whenever --
((P_{t }+ G_{t})
- (P_{t-1 }+ G_{t-1})) =
D(P_{t-1 }+ G_{t-1}) =
DS_{t-1 } > 0
D(P_{t-1 }+ G_{t-1}) =
DS_{t-1 } > 0
-- and that positive ‘societal
[self-re]productive [self-]acceleration’/‘societal [self-re]productive [self-]force’ exists, for time-period
t,
whenever there is a gain in
positive-signed ‘self-[reproductive] momentum’ in time-period t, relative to time-period t-1
--
((P_{t }+ G_{t})
- (P_{t-1 }+ G_{t-1})) >
((P_{t-2 }+ G_{t-2}) - (P_{t-1 }+ G_{t-1}))
-- i.e., whenever --
D(P_{t-1 }+ G_{t-1}) > D(P_{t-2 }+ G_{t-2}))
-- or whenever --
DS_{t-1} > DS_{t-2}.
((P_{t-2 }+ G_{t-2}) - (P_{t-1 }+ G_{t-1}))
-- i.e., whenever --
D(P_{t-1 }+ G_{t-1}) > D(P_{t-2 }+ G_{t-2}))
-- or whenever --
DS_{t-1} > DS_{t-2}.
In the image below, the ‘sgn((P_{t-2}_{ }+ G_{t-2}) - (P_{t-1}_{ }+ G_{t-1}))’ factor in the numerator serves to rectify the sign of the whole expression, when either the sign of only the
((P_{t-2}_{ }+ G_{t-2}) - (P_{t-1}_{ }+ G_{t-1})) denominator "goes minus", or when the signs of both the ((P_{t }+ G_{t}) - (P_{t-1 }+ G_{t-1})) numerator-factor, & that denominator, both “go minus” together.
We noted, above, that
“the variables G_{t} and P_{t} are not defined for negative values.”
¿But what happens if G_{t}, or P_{t}, or
both, become 0,
for any time-period t?
For the ‘human-societal
self-reproductive self-force’ to even exist -- indeed, for
the human species,
as such, to even exist, for long -- both P_{t}
and G_{t} must be greater than
zero.
Living representatives
of the human Genome, but stripped of all ‘Phenomic artifacts’, and so
remaining, do not constitute a viable, sustainable
human-societal unit, and therefore do not qualify as manifesting the
existence of human-societal
self-reproductive force.
An accumulation of
‘Phenomic artifacts’ alone, no matter how massive and how vast, if devoid of any
living human population, do not constitute a human-societal unit at all, and, likewise, qualify even
less as manifesting the existence of human-societal-reproductive self-force.
In either case, the value of this ‘self-force’ is not ‘0’, not the "purely" quantitative 'empty zero' value, but, rather, takes on the 'quanto-qualitative' ‘‘‘value’’’ ‘non-existent’, connoted by the 'full-zero' symbol, ‘.’.
Therefore, the factor (P_{t }× G_{t})/(P_{t }× G_{t}) in the image of the productive-force formula below is included, so that the formula will generate the 'quanto-qualitative' [e]valu[e][ation] ‘non-existent’, ‘.’, in any time-period for which the G_{t} quantifier by itself goes to the "purely" quantitative 'empty zero' value, 0, or in any time-period for which the P_{t} quantifier by itself goes to the "purely" quantitative 'empty zero' value, 0, or in which both the P_{t} quantifer and the G_{t }quantifier, together, go to the "purely" quantitative 'empty zero' value, 0.
The value of this (P_{t}G_{t})/(P_{t}G_{t}) factor is 1, producing no impact at all on the overall value of the metric, as long as both the P_{t} quantifier and G_{t} quantifier are greater than 0.
But if either the quantifier P_{t} or the quantifier G_{t}, or both, go the quantifier value ‘0’, which Seldon calls ‘empty zero’, then we have a ‘‘‘division-by-zero singularity’’’, although, in this case, it is ‘empty zero’ itself that is being divided by ‘empty zero’ itself, not, as more typically for singularities, some non-0 quantifier value that is being divided by the 'empty zero' the quantifier, 0.
The value of the ‘quantifier’ part of the whole metric thus becomes (0/0), which is an "indeterminate value", an inherently non-specific value -- an "unsolvable" numerical value syntax/symbol, an arithmetical symbol which cannot be "solved" as any specific value, or as any specific '''solution subset''' of values -- within the set of the so-called "Real" numbers", R.
Taking into account also the ‘qualifier’ part of the whole metric, i.e., the ‘metrical qualifiers’ that represent the “grams” metrical unit, denoted by ‘gm.’, we have ‘full zero’, as the overall result, whenever P_{t}G_{t} = 0, for the entire metric expression [note, herein, by the term 'metrical', we mean 'measuremental'; we do not necessarily mean only the so-called "metric system" of dimensional units, i.e., we do not restrict the dimensional unit qualifiers so referenced to, e.g., ‘sec.’, ‘gm.’, ‘cm.’, etc.] --
(0)[gm.]/(0)[gm.] |- = . /. |- = .
-- wherein we have used the ideogramic relation symbol ‘|- =’ to stand for the phrase 'is equal to by definition' and 'is equal to by axiomatic assertion'.
The image immediately below
illustrates the result of our construction of this latest candidate metric for the Marxian, human ‘‘‘social
[re-]productive force’’’
--
That is, when, in the seventh Seldonian dialectical arithmetic/algebra, a ‘‘‘quantifier’’’ value of ‘empty zero’, 0, multiplies the ‘metrical qualifier’ in one of its ‘qualo-quantitative’ arithmetical /- algebraic expressions, the result is axiomatically, but also intuitively, held to be what Seldon calls ‘full zero’, which we denote by ‘.’ -- a symbol which stands for a new kind of arithmetical value, one which first takes full shape in this Seldonian seventh system of dialectical arithmetic.
This ‘full zero’ value, in the axioms-system of that seventh Seldonian dialectical arithmetic, stands for ‘no longer exists’, i.e., for the qualitative, ontological ‘‘‘non-existence’’’ of the entity measured by the metric formula when that metric formula takes on this ‘.’ value, signifying, in the metric formula of the image above, as in the logic of our argument, above, that human-social productive force can exist only while both P_{t} and G_{t} maintain above 0 values, i.e., while --
P_{t} > 0 < G_{t}.
The meaning of the value named "full zero". The 'meta-number' denoted by ‘.’ does not signify contra-empirical states of affairs, such as '''actual infinity''', or '''absolute nothingness'''.
When a metric, or an equation, takes on the value ‘.’, this does not signify "abstract negation".
On the contrary, the value ‘.’ represents that state of affairs in which a given mathematical model has breached the limits of its descriptive capacity.
This state of affairs, denoted ‘.’, in a dynamical system, is analogous to Godel incompleteness in an axiomatic system of arithmetic, wherein the statement of an "unprovable theorem" of that system "deformalizes" into a system-specific ["diophantine"] equation whose specific solution cannot be expressed in terms of the '''arithmetical vocabulary''', or '''numerical vocabulary''', that is available in that axiomatic system of arithmetic.
The value ‘.’, represents a situation in which, as predicted by the given dynamical equation or model, an older, concrete, specific, determinate state of affairs -- a state which CAN be described within the limitations of the "language" of that model -- gives way to, or turns itself into, a new, concrete, specific, determinate state of affairs -- a state which can NO LONGER be described within the limitations of the "language" of that model -- one which CANNOT be expressed given the '''ontological commitments''' of the model specification, its '''vocabulary''' and '''terms of reference''', BECAUSE some of the old ontology presupposed in that model specification has ceased to exist, has descended into non-existence, and/or in which some new, e.g., unprecedented ontology, not referenced in that model specification, has arisen into existence, e.g., for the first time.
[For the 'social reproductive force' metric presented herein, examples might include, for some future time-period, t*, a period 'characterizeable' as one of 'metafinite singularity' for the given human-social unit, the disappearance of the entire human-Phenomic social infrastructure as such, as signified by P_{t*} = 0, coupled with the emergence, for the first time, from out of the most concentrated core of the given human civilization, of the unprecedented 'new ontology' of a '''meta-human''' population of '"artificially-intelligent''', e.g., of '''android''', robots, interconnected with a disappearance, within that social unit, of the living human population, as signified by G_{t*} = 0.
Alternatively, examples might include a future total assimilation of the entire living, Genomic human population of a given social unit, as signified by G_{t*} = 0, and of its entire, Phenomic social infrastructure, as signified by P_{t*} = 0, into/by a '''meta-human''' population of '''cyborgs''', combining both human-Genomic features, including a technologically-reengineered, formerly-human, Genome, as signified by G_{t*} = 0, and robotic features, so that --
P_{t ---> t*} ---> 0 <--- G_{t ---> t*}].
The sign ‘.’ thus signifies the outcome of a "determinate negation" of a former state of affairs, not of an "abstract negation" [e.g., the transition of that concrete state of affairs into "absolute nothingness"], but of a "determinate negation" which cannot be described "determinately" -- concretely and specifically -- within the "mathematical language" limitations of the model specification, and which thus can only be described, within the limitations of that model specification, generically, by the generic ‘.’ value, because the specific "determinations" of the new state of affairs -- the result of the "determinate [self-]negation" of the former state of affairs -- are beyond the expressive capabilities of the model which produced this ‘.’ value as its prediction for a time beyond it descriptive reach; a model which generated this ‘.’ value as a sign of its limited expressive/descriptive capabilities reaching, and breaching, their limits.
Of course, this
productive force
metric can momentaneously and transiently pass through the ‘full zero’ value, and state,
whenever the velocity of social
reproduction transiently crosses over, from negative to positive values, or
from positive to negative values, momentarily ‘demanifesting’ this force [e.g., during transitions into, or out of, those social states of famine, economic
depression, war, genocide, "dark ages", etc., which are characterized by negative-signed rates of human social reproduction, i.e., by states of contracting human social reproduction].
But G_{t} = 0 and/or P_{t }=
0 means lasting ‘demanifestation’ of this force for the
social locus so-measured.
For more about this seventh of the Seldonian dialectical arithmetics, the “Mu” arithmetic, _{R}mu = _{R}q_{M}_{U} = _{R}q_{M}_{Q}_{N} [----) q_{7},
and its new ‘meta-number’, ‘full zero’, ‘.’, see pp. A-9 to A-14,
and pp. A-18 to A-21,
in Supplement A
to the F.E.D. Introductory Letter, composed by the F.E.D.
General Council’s Secretary-General, Hermes de Nemores --
http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf .
For more about this seventh of the Seldonian dialectical arithmetics, the “Mu” arithmetic, _{R}
http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf .
TO
BE CONTINUED.